A scaling characterization of nc-rank via unbounded gradient flow

Abstract

Given a tuple of $n\times n$ complex matrices $A=(A_1,A_2,\dots ,A_m)$, the linear symbolic matrix $A=A_1{x}_1+A_2{x}_2+\cdots +A_m{x}_m$ is nonsingular in the noncommutative sense if and only if the completely positive operators $T_A\left(X\right)={\sum }_{i=1}^m{A}_{i}X{A}_i^{†}$ and $T_A^{⁎}\left(X\right)={\sum }_{i=1}^m{A}_i^{†}X{A}_i$ can be scaled to be doubly stochastic: For every $ϵ>0$ there are $g,h\in GL(n,C)$ such that $\Vert T_{g^{†}Ah}\left(I\right)-I\Vert <ϵ$, $\Vert T_{g^{†}Ah}^{⁎}\left(I\right)-I\Vert <ϵ$. In this paper, we show a refinement: The noncommutative corank of A is equal to one-half of the minimum residual ${\Vert T_{g^{†}Ah}\left(I\right)-I\Vert }_1+{\Vert T_{g^{†}Ah}^{⁎}\left(I\right)-I\Vert }_1$ over all possible scalings $g^{†}Ah$, where ${\Vert \cdot \Vert }_1$ is the trace norm. To show this, we interpret the residuals as gradients of a convex function on the symmetric space $GL(n,C)/U_n$, and establish a general duality relation of the minimum gradient norm of a lower-unbounded convex function f on $GL(n,C)/U_n$ with an invariant Finsler metric, by utilizing the unbounded gradient flow of f at infinity.